CC BY
9
УДК 517.9
On Local Solvability of the System of the Equations of One Dimensional Motion of Magma
Alexander A. Papin* Margarita A. Tokareva^
Altai State University Lenina, 61, Barnaul, 656049
Russia
Received 01.12.2016, received in revised form 20.01.2017, accepted 20.05.2017 The local solvability of initial-boundary value problem for the system of the equations of non stationary motion of magma is proved.
Keywords: Darcy law, poroelastisity, magma, solvability, uniqueness. DOI: 10.17516/1997-1397-2017-10-3-385-395.
1. Problem statement. Formulation of main results
A quasi-linear system of equations of composite type is considered:
^ + £((i - = 0 ^ + d<-">+>> = »• «)
$(vf - vs) = -k($)(df - pfg), (2)
dvs 1 , .
sPe, Pe = Ptot - Pf, (3)
dx m
= -Ptotg, Ptot = $Pf + (1 - $)Ps, Ptot = $Pf + (1 - $)Ps. (4)
We seek a solution of this system in the domain (x,t) e QT = ^ x (0, T), Q = (0,1), under the boundary and initial conditions
Vs \x=0,x=i= Vf |x=0,x=i=0, $ |t=0= $0(x), Pf |t=0= P0(x). (5)
This quasi-linear system of equations describes 1D non-stationary isothermal motion of magma in porous rock. The laws of conservation of mass for each phase, Darcy's law for fluid phase, taking into account the motion of a solid skeleton, the rheological law and the equation of conservation of momentum for system describe this process [1-3]. Here Pf, Ps, Vf, vs are, respectively, real density and velocity of solid and fluid phases, $ is the porosity, Pf ,Ps are, respectively, pressures of the fluid and solid phases; Pe is the effective pressure, Ptot is the total pressure, Ptot is the density of the two-phase medium, g is the density of the mass forces; k($) is the coefficient of filtration, £($) is the coefficient of rock shear viscosity (specified function).
1 [email protected] © Siberian Federal University. All rights reserved
The problem is written in the Eulerian coordinates x, t. The real density of the solid particles ps is assumed constant. The unknown quantities are pf, vf, vs pf, ps. The system of equations (1)-(4) is closed either by using the equation of state of the fluid phase p f = p(pf) (in particular,
dp f 1
the relationship may be, commonly used in applications = —-, where ¡f is the fluid
dpf fpf
compressibility [1-3]).
The numerical studies of various initial boundary-value problems for systems of equations (1)-(4) were carried out in [2, 3]. Some exact solutions have been constructed in [4]. In these studies the following dependencies of the functional parameters of the problem was used: k($) = k$n/p, 1/£($) = /v, where m G [0, 2], n = 3; v, p, k are positive environment settings [2].
Structurally similar systems of equations was considered in [5-7]. In these studies, based on a number of simplifying assumptions, the original system were reduced to one higher order equation. The local solvability of the Cauchy problem in Sobolev spaces was established in [5]. Travelling wave solutions have been studied in [6,7].
In this paper the unique local solvability of problem (1)-(5) is proved in the case when g = 0 and pf is function of pressure.
On Q and QT, let us consider several function spaces, using the notation from [8]. Suppose that || • H^n is the norm on the Lebesgue space Lq(Q), q G [1, to]. For brevity, let || • ||q = || • ||q,o, II • II = || • ||2,n .We also use the Holder spaces Ca(Q), Ck+a(Q), where k is a natural number and a G (0,1] with norms:
|c*(n) = f U,n = f |o,n + Hxa(f), f |o,n = max f (x)|,
xeQ
Ha(f ) = sup f (xi) - f (x2)||xi - X2^a,
Xi ,X2 En
f ||cfc+„(n) = fk+an = it UDZf||o,n + Ha(DXf).
m=0
For functions given on QT, we need the space Ck+a'm+l3(QT), where k, m are natural num-
k
bers and (a,3) G (0,1], with norm HfHck+am+e(qt) = f = E HD'Xf||o,Qt +
1=0
tt Df ||o,QT + Ha(Dkf)+ +Hf(Dkf) + Ha(Dmf) + H(D?f), where
j=i
Ha(f (x,t))= sup f (xi,t) - f (x2,t)||xi - x2-a, x1, x 2en,te(o,T)
H (f (x,t))= sup f (x,ti) - f (x,t 2)||ti - t2 —.
t1 ,t2E(o,T ),xEn
In the case k = m and a = 3, we use the notation Ck+a (QT).
In this paper by a solution of problem (1)-(5) we mean the set of functions vs G C3+a'i+a/2(QT) ($, Pf, pf, ps) G C2+a'i+a/2(QT), vf G C i+a'i+a/2 (Qt ), such that 0 <$< 1, Pf > 0, pf > 0. These functions satisfy the equations (1)-(4) and the initial and boundary conditions (5) and regarded as continuous functions in Qt . Let us state the main results of the paper.
Theorem 1. Suppose that g = 0 and the data of problem (1)-(5) satisfies the following conditions:
1) the functions k($),£($),pf (pf) and their derivatives up to the second order are continuous for $ G (0,1), Pf > 0, and satisfy the conditions
k^r1 (1 - $)q2 < k($) < ko$q3 (1 - $)q4, !/£($) = ao($)$ai (1 - $)a2-i, 0 <Ri < ao($) < R2, k^pf < pf (Pf) < kopf, k^pf < < kopf,
where k°, ai: R^,i = 1, 2 are positive constants, q1, ...,q8 are fixed real parameters;
2) the initial functions 4°, P°satisfy the following smoothness conditions: 4° G C"2+a(Q),p0 G
C2+a(il) and the matching conditions
dPf (P°)\ = 0
-;- \x=0,x=1 = °
dx
as well as satisfy the inequalities
0 <m0 < 4°(x) < M° < 1, 0 <m1 < p°(x) < M1 < x>, x G Q,
where m°, M0,m1, M1 are given positive constants. Then problem (1)-(5) has a local solution,
1.e., there exists a value of t° G (0,T) such that
va(x,t) G C3+a'1+a/2(Qt0), (4(x,t),Ps(x,t),Pf (x,t),pf (x,t)) G C2+a'1+a/2(Qto),
Vf (x,t) G C 1+a'1+a/2(Qt0). Moreover, 0 < 4(x,t) < 1, Pf (x,t) > 0 e Qt0.
2. Local solvability
Under the conditions of the theorem into force (4) we have ptot = p°(t). Following [9], we rewrite the system (1)-(3). Suppose that x = x(r,x,t) is a solution of the Cauchy problem
dx
— = Va(x,T), x \T=t= x. or
We set x = x(0, x, t) and take x and t for the new variables. Then 1 — 4(x, t) = (1 — 4°(x))J(x, t),
~ dx
where J(x,t) = — (x,t) is the Jacobian of the transformation. The system of equations (1)-(3) in the new variables is of the form
d(l — 4) + 0—41 dV a 0 d (, r,+ (±—4) 9 r 4 -M (! — 4) d (* 4)
~dT- + -Y-4rdx = 0 dt(pf4) + T—4O dx(pf^) = va 1—4°dx(pf4),
VJ" " \ — 4) dpf (1 — 4) dVa
4 a — V f ) = k(4)T—W ~d£, dx = —ai(4)pe,
where a1 (4) = 1/^(4)-Since
-,dvs
Vsdx(pf4) = dx(pf4vs) -pf4dx,
it follows that the continuity equation for the liquid phase can be reduced to the form 1 d * 1 d * 1 „ *dva
(1—4) di(Vf 4) + 1—4 dx(V f 4(v f — va + 1—4V f 4dx =
Using the continuity equation for the solid phase, we find that
it (Vf ) + j—n i{Vf 4(v f — va)) =0.
Finally, passing from (x, t) to the mass Lagrangian variables (y, t) by the rule
,-x
(1 — 4°(x))dx = dy, y(x)= (1 — 4°(n))dn G [0,1]
J 0
and preserving the notation y for the variable x, we obtain
- %=»■ !(« î^)+ix p «"> -*»=<>.
- vf )=- dx,
(1 - ^ df = ^i^^ Pe = p0 (t) - Pf . Finally, we turn to the dimensionless variables
t , X , vs , Vf , Pf
t= —, X=-, vs = —, Vf = , Pf = , ti L Vi ' Vi ' Ps
Pf = f, PS = ^, P>e = *, ^ = PpO, ai(^) = , k' (*) = M,
' Pi Pi Pi Pi a0 ki
— , a° = ^ lo vi lpi
having the dimension of velocity and pressure accordingly.
Then the domain x' is the unit interval [0,1] and the system of equations will retain its
structure (dashes omitted).
Using the rheological relationship, Darcy's law and the conditions vs|œ=0ji = 0, we find that
fi L v v L where L = (1 — 4>0(v))dn, ti = —, a0 = ——, ki =-, vi are fixed positive quantities
Jo v i Lp i p i
p°(t)=i i—¡Pfdx(o a—)x ^ ^(t,pf).
ai ($) p dx II "iy^! dx\ ^ po( Taking into account Darcy's law, the second equation of the system assumes the form
Upf 1—j) — Í(P> k<*)(i — « f =»■
From the first and fourth equations of the system follows that
ai($)(pf - po)■
1 W _ o,
1 — <pot
This equation can be written as
3G(0)
dt = Pf — P where the function G(^) is defined by the equation
dG(^) _ 1
0
dfi (1 — t)a i (t)
Let
a(^) = î——-, K (t) = k(t)(1 - t), b(pf ) = pf ff.
Taking into account the conditions (5), we obtain the following problem for finding functions pf ,$:
Ît (a(^f ) - KK(^)b(Pf) d~B) =0, (6)
Pf (pf ) - P0(t), (7)
df lx=0,x= i =0, Pf |t=0 = P0(x), t |t=0= 4>°(x)- (8)
Lemma 1. Let the data of problem (6)-(8) satisfy the conditions of the theorem. Then problem (6)-(8) has a unique local solution, i.e., there exists a value of t0 such that
(4(x,t),Pf (x,t)) G C2+a'1+a/2(Qt0). Furthermore, 0 < 4(x,t) < 1, pf (x,t) > 0 in Qt .
The solvability of problem (6)-(8) is established by using the Tikhonov- Schauder fixed-point theorem: if V is a compact convex closed set of Banach space B and the operator A maps V into itself continuously in the norm of B, then there is a fixed point on V [10, pp. 227].
Since the function 4 = G(4) is strictly monotone, at 4 G (0,1), that the inverse function is exist: 4 = G-1(4). Assuming that p(x,t) = pf (x,t) — p0(x), u(x,t) = G(4) — G(40). We represent the equations (6),(7) in the form
I «")(p+^ = ¿(K o)b(p+p0) '-^dxr1), (9)
^ = Pf (p + p0) — p0(t). (10)
Here a(u) = 4(^ , K(u) = k(4(u))(1 — 4(u)), 4(u) = G-1(u + G(40)). Moreover,
1 - 4(w)
«,=== U„, = »■ <">
For the Banach space, we choose the space C2+l3'1+l3/2(Qto), where / is any number from the interval (0, a), a G [0,1). Let
= 0,
x=0,x=l
V = {p(x,t),u(x,t) G C2+a'1+a/2(Qt0)| p \t=0= u |t=0= dx — p0(x) < p(x, t) < 2M1 — p0(x) < x, G(m0/2) — G(40) < u(x,t) < G(— G(40) < x, (x,t) G Qto,
(|u\1+a,(1+a)/2,Qt0 , \p\1+a,(1+a)/2,Qt0 ) < K1, (|u\2+a,(2+a)/2,Qt0 , \p\2+a,(2+a)/2,Qt0 ) < K1 + K2 } , where K1 is an arbitrary positive constant, while the positive constant K2 will be given later. We note that on the set V following inequalities hold: 0 < m2° ^ 4(u) < M°2+ 1 < 1, -(u) > 0, K(u) > 0.
Let us construct an operator A mapping V in V. Suppose that u,p G V. Using (10), we define the function u by the equality
u =/Tpf (P(x,T) + p0(x)) —jf T—^Pf (P(x,T)+ p0(x))dx[Jo1 dx) ) T (12)
From the representation (12) it follows that smoothness u is determined by the smoothness of functions p,p0 and p0. In particular, we have an estimate
\u\2+a,1+a/2,Qt0 = C1(m0 ,M0,m1,M1,K1,T, \p0|2+a,n)(1 + ^ Pxx\a,a/2,Q).
Lemma 2. Let function ai($), $ G (0,1) satisfies the following condition
(1 - $)ai($) = ao($)$ai (1 - $)a2, 0 <Ri < ao($) < R2,
where Ri > 0, ai > 0, i = 1, 2. Then we have the estimate of the form
R2G($i) - G($2)| > $i - $2\-
Proof. Assume without loss of generality that 0 < $i < $2 < 1. From the definition of functions G($) and ai($), we have
r^2 ds 1
0 < AG = G($2) - G($i) = --—- > —($2 - $i).
U1 (1 - s)ai(s) R2
Lemma 2 is proved. □
In this way, we have estimate
^(x^) - $0(x)| < S(t), S(t) —> 0 as t —> 0,
which implies, that there exists a value ti = ti(mo,Mo,mi,Mi), such that for all to < ti the following inequality holds
0 <m < $(x,t) < Mo2+1, (x,t) G Qto ■ (13)
Taking into account (13) we also have the estimate for function w(x,t): G ^ w(x,t) +
„( Mo + 1
G(t0) < G ( 2
Using (9), (11) and 0j(x,t) we find the function p(x,t) as a solution of the problem (here and elsewhere, we assume that the initial and boundary conditions are matched):
I (a(S>0 + P0)) = ¿(K (^
(14)
P I = dpf
p ^0= -ex
dp0
= 0, dx
= 0.
x=0,x=i
The equation for p(x,t) is uniformly parabolic. In view of the properties of w(x, t) and po(x) problem (14) has a classical solution [8]. In addition, we have the following estimate:
1 3a(w)
a(w) dt
< C0(m0,M0,mi,Mi, max |P°(t)|).
0<t<T
Under the additional condition smallness for the value of the time interval the following statement holds [9].
Lemma 3. For to < min(ti,t2), t2 = ln2/Co(mo, Mo,mi ,Mi), the classical solution of problem (14) satisfies the following inequality in Qt0 :
0 <m2i < p(x, t) + po(x) < 2Mi < to.
Proof. Further, setting U(x,t) = p(x,t) + po(x), we can express problem (14) in the form d , , d / , ,,, ,dU\ dU
_ p 0 t=0 "
x=0,x=1
U|t=0 _ p0. (15)
First, we show that U(x,t) > 0, (x,t) G Qt0. In equation (15), let us make the change U(x,t) = -z(x,t). Then
da + dz d Kfrdz) dt dt dx dx
Let
z(o)(x,t) = max{z,0}, z(o)(x,t) |t=o= max{-po,0} = 0,
ae(x,t)= z^^^t)^^,^2 + £)-i/2, £> 0.
Let us multiply the equation for the function z by aE and then integrate over Q. We obtain the equality
dt 0 a(\z(0)\2 + e)1/2dx + 0 dt- (|z(0)|2 + e)1/2)dx+
(16)
+e f1 Kbdz^(z^f + e)-3/2dx = 0. Jo dx dx
Let A+(t) = {x G Q| z(x,t) > 0}, A-(t) = {x G Q| z(x,t) < 0}. Then
£ dt(zaE - (z^^ + e)i/2)dx = -e J ^(M2 + e)-i/2dx - ei/2 j ^dx, o A+(t) A-(t)
/ a(\z(0)|2 + e)i/2 dx = / a(|z| + e )i/2d x + e}/2 J adx,
o A+(t) A- (t)
/ adz^f + e)i/2\t=0dx = exl2\ a\t=odx, a^2 + e)i/2dx ^ a^dx = / az(0)dx. o t=o o t=o o
A+(t) A+(t)
Integrating relation (16) with respect to time, we obtain
J a(\z\2 + e)1/2dx + e1/2 J adx + ej J Kb\^\2(z2 + e)-3/2dxdr i+(t) A-(t) 0 A+(r)
= e 0 f ^(\z\2 + e)-1/2dxdr + e1/2 j* j ^dxdr + e1/2 £ a \t=o o
to j u> JO
A+(T ) A- (T )
Therefore,
i az(0)dx < £i/2 i i ^ dxdr + £i/2 i a\ dx.
Jo Jo Jo dT Jo =
Passing to the limit as e ^ 0, we find that z(o) = 0, i.e. U ^ 0.
After multiplication by U1-1 (x,t), l > 2, equation (15) can be expressed as
—->+(i - ^ ^ f )2+- u ' da=i f).
l dt
t
Then
Therefore,
fi
1 — I aUdx ^ l dt J o
l - 1
—;— max
l 0<œ<1
1 da a dt
i
aUl dx.
l-1
y (t ) ^ —;— max
1 da a dt
l Ox^xi
I
After passing to the limit as l ^ œ, we obtain
y(t) < y(0) exp <j 0 max
y(t), y1 (t) = f (a1/lU)ldx, o
o
1 da a dt
max U(x,t) ^ max p°(x)exp^ [ max \ — — \dr|.
OXœXl
OXœXl
o
Taking into account the inequality max pp (x) ^ Mi and choosing t from the condition
OXœXl
a:
i ft2 ,1 dat , 1
t < t2, exp < max h — \dr > < 2, 1 ' ox®Xi a dt j
we obtain upper bound for p. To obtain a lower estimate we represent equation (15) in the form
(z(x,t) = 1/U (x,t))
1ii(wl+c+DKbz'-2( D2—^ z' da=d (Kbz-1 t)
Then we obtain inequality
1 d Idt,
then we obtain the estimate
1
max
ldx <
l +1
—;— max
l ox®Xi
< max
OXœXi U (x,t) OXxX. i p0(x)
exp
u:
1 da a dt
max
OXœXi
1 da a dt
dr\ < —.
I mi
Lemma 3 is proved. □
In view of Lemma 3 and the properties of u, we have the following estimates [8, Sec. 3]:
\p\a,a/2,Qt0 < C2,
\p\2+a,1+a/2,Qt0 ^ C3 + \p0\2+u,Q, + \px\a,a/2,Qt0 + \ut\a,a/2,Qt0 + \ux\a,a/2,Qt0^ , in which the constant C2,C3 depends on K1, m0, m1, M0, M1. Therefore
\p\2+a,1+a/2,Qt0 < CA(K1, m0, m1, M0, M1).
K1 + K2
Let C5 = max{C1,C^\. Choose K2 so that C5 < ———1. Then, for t0 < min(t1,t2, (K1 + K2)-1) we obtain
\p\2+a,i+a/2,Qt0 < Ki + K2, \u\2+a,i+a/2,Qt0 < Ki + K2.
O
i
1
az
az
O
O
1
It remains to verify conditions
\p\i+a,(i+a)/2,Qt0 < Ki, \w\i+a,(i+a)/2,Qt0 < Ki.
Integrating equation (14) with respect to time, we obtain \p\oQQto < C6t0. From the equation (12) we obtain \w\0Qto < C7t0. Further, using for p,w an inequality of the form [11, pp.35]
\U\i+a,(i+a)/2,Qt0 < C\u\C2+a,1+a/2,Qt0 MoQ, , c = (1 + a)(2 + a)~i,
we find that there exists a sufficiently small value of t0, depending on K1 and K2, such that the required estimates hold: \p\i+a(i+a)/2,Qt0 < Ki, \^\i+a,(i+a)/2,Qt0 < Ki.
Thus, the operator A maps the set V into itself for sufficiently small values of t0. Using the estimates obtained above, we can easily show the continuity of the operator A in the norm of the space C2+l'i+l/2(Qt0). By the Tikhonov-Schauder theorem, there exists a fixed point (p, w) € V of the operator A.
Let us establish uniqueness of the solution of problem (6)-(8).
Suppose that (pf,4>(i)) and (pf,4>(2)) are two different solutions of problem. Their difference p = pf — pf,t = j>(i) — 4>(2) is the solution of the linear homogeneous system
d Of dp \
— (dop + di4>) - -Q^id"2dx + d3P + \ =0, (17)
d
— (hot) - hip + y(t) = 0, (18)
f1
y(t) = P°(t(1\pf) - P°(t(2\pf) = J (h2(x,t)p(x,t) + h3(x,t)t(x,t))dx,
with zero initial and boundary conditions $ |t=o= P |t=o= The coefficients
„(2)
dp
dx
= 0.
x=0,x=1
/ mx (a(t(1)) - a(t(2)))Pf ( (2)) ( (2))
do = a(t(1)) > 0, di = V V ^ - f > 0, d2 = K{t(2)) b{pf) > 0,
«) _ b(P(2)) dp(1) K(t(1)) Kt±(2)) dp(l)
(2))
,, = k^ ^^P- f d = f - ^ ^
G(t(1)) - G(t(2)) n , _P(P f) - P(P f)
h0 = t(1) - № > 0 h1 = pf - pf
f --■ f \i: ^dx}-\
^^ = a1(t(1)) Pf (pf ) - Pf (pf ) í Í1 a1(t(1)) dx
1 - t(1) pf - pf \Jo 1 - t(1)
h3=( rS - 0-SO <*m -1»)
1
y
1 -1(1) 1 -1(2)"" " ' -1 „ 1 ( ^^ , „1 , „ 1 , x - 1N
{f pm o am df n ^ >» (f>H s: ■m dx o ^-y)
are bounded for all x G [0,1], t G [0,T],
x
Taking into account (18), equation (17) can be represented as
it (dop) + ho (hip- V (t))+ho$it{io)- i{d2 dx+d3 p+d$ =
Multiplying the equation (19) by p(x,t) and integrating by x from 0 to 1, we obtain
dt J p\(x,t)dx < C^ J p2(x,t)dx + J u2(x,t)dx + V2(t)^j ,
(19)
(20)
i/2
where pi(x,t) = do ^(x^)^ u(x,t) = h0$(x,t). Here the constant C depends on T and quantities
1
max ... „--, max -Tt---, max pf)(x,t), max -^rv
t)eQr $(i)(x,t) (x,t)eQr 1 - $(i)(x,t) (x,t)eQT f (x,t)EQT p(i)
(x,t)EQT $(i)(x,t) ( d$(i)(x,t)
max
(x,t)EQT
dt
max
(x,t)EQT
3pf(x,t)
dt
max
(x,t)EQT
Opf(x,t)
dx
pf (x,t)
i = 1, 2.
For V(t), we also have V(t) < C / (pi(x,t) + |u(x,t)|)dx■
o
Integrating equation (18) by time and taking into account the estimate for V(t), we obtain |u(x,t)| ^ C/ (pi(x, r) + |V(r)Q dr ^ C / pi(x,r)dr+ / pi(x,r)dxdr+ / |u(x,т)|dxdт).
Jo \Jo Jo Jo Jo Jo J
I ntegrating last inequality by x from 0 to 1, we obtain Gronwall inequality for function
| u(x, t) | dx:
\u(x
t)\dx ^ / Pi(x,T)dxdr + / \u(x,T)\dxdr .
o o o o
Therefore
r-l
/ |u(x,t)|dx ^ C / pi(x,T)dxdT, [V(t)| ^ C / pi(x,t)dx + / pi(x,T)dxdT ,
Jo Jo Jo \Jo Jo Jo J
and consequently |u(x,t)| < C^ J pi(x,T)dT + J J pi(x,T)dxd^j. Hence we obtain from
(20): d 0 ( 0 0t )
Jt \\pi(t)f < C(\\pi(t)f + ^ \\pi(T )\\2dTj. (21)
We set w(t) = j \piO\dT, then from (21) we obtain dw < + w(t)) ■ This yields
^(e^dw - (C + 1)w^ + wet < 0, so we have inequality ^ (C + 1)w. Therefore w(t) = 0
d j tfdw dt\e \~dt
h p = 0, $ = 0. Lemma 1 is proved.
After finding $ and pf, we find ptot = Po(pf, $). Then we find ps = (po — $pf )(1 — $)-1. We dv
find vs from the equation = —a1($)(1 — $)-1(ptot — pf), and from the Darcy's law we obtain
dx
Vf = Vs — k($)(1 — $)$-1 dpf.
1
o
1
o
Since (ф,р{) e C2+a>1+ 2 (Qt0), then we have: va e C3+a'1+2 (Qt0), vf G C 1+a'1+ a (Qt0), (pf ,pa) G C2+a'1+a (Qto). □
This work was partially supported by the grants RFBR 16-08-00291 "Hydroelastic and thermodynamic effects with inter-action of poroelastic ice and structures" and state assignment of the Ministry of Education and Science no. 2014/2.
References
[1] J.Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.
[2] J.A.D.Connolly, Y.Y.Podladchikov, Compaction-driven fluid flow in viscoelastic rock, Geodin. Acta, 11(1998), 55-84.
[3] C.Morency, R.S.Huismans, C.Beaumont, P.Fullsack, A numerical model for coupled fluid flow and matrix deformation with applications to disequilibrium compaction and delta stability, Journal of Geophysical Research, 112(2007), B10407.
[4] I.G.Akhmerova, A.A.Papin, M.A.Tokareva, Mathematical models of heterogeneous media. Part 1., Altai Gos. Univ., Barnaul, 2012 (in Russian).
[5] M.Simpson, M.Spiegelman, M.I.Weinstein, Degenerate dispersive equations arising in the stady of magma dynamics, Nonlinearity, 20(2007), 21-49.
[6] A.M.Abourabia, K.M.Hassan, A.M.Morad, Analytical solutions of the magma equations for molten rocks in a granular matrix, Chaos Solutions Fract., 42(2009), 1170-1180.
[7] Y.Geng, L.Zhang, Bifurcations of traveling wave solutions for the magma equation, Applied Mathematics and computation, 217(2010), 1741-1748.
[8] O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Moscow, Nauka, 1967 (in Russian).
[9] A.A.Papin, I.G.Akhmerova, Solvability of the system of equations of one-dimensional motion of a heat-conducting two-phase mixture, Mathematical Notes, 87(2010), no. 2, 230-243.
[10] R.Edwards, Functional Analysis: Theory and Applications, New York, 1965.
[11] S.N.Antontsev, A.V.Kazhikhov, V.N.Monakhov, Boundary-Value Problems of the Mechanics of Inhomogeneous Fluids, Nauka, Sibirskoe Otdelenie, Novosibirsk, 1983 (in Russian).
Локальная разрешимость системы уравнений одномерного движения магмы
Александр А. Папин Маргарита А. Токарева
Алтайский государственный университет Ленина, 61, Барнаул, 656049 Россия
Для системы уравнений одномерного нестационарного движения магмы доказана однозначная локальная разрешимость начально-краевой задачи.
Ключевые слова: закон Дарси, пороупругость, магма, разрешимость, единственность.
